Average Error: 0.5 → 0.5
Time: 4.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)\right)}\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)\right)}\right)\right)
double f(double v) {
        double r235154 = 1.0;
        double r235155 = 5.0;
        double r235156 = v;
        double r235157 = r235156 * r235156;
        double r235158 = r235155 * r235157;
        double r235159 = r235154 - r235158;
        double r235160 = r235157 - r235154;
        double r235161 = r235159 / r235160;
        double r235162 = acos(r235161);
        return r235162;
}
↓
double f(double v) {
        double r235163 = 1.0;
        double r235164 = 5.0;
        double r235165 = v;
        double r235166 = r235165 * r235165;
        double r235167 = r235164 * r235166;
        double r235168 = r235163 - r235167;
        double r235169 = 3.0;
        double r235170 = pow(r235166, r235169);
        double r235171 = pow(r235163, r235169);
        double r235172 = r235170 - r235171;
        double r235173 = r235168 / r235172;
        double r235174 = r235166 * r235166;
        double r235175 = r235163 * r235163;
        double r235176 = r235166 * r235163;
        double r235177 = r235175 + r235176;
        double r235178 = r235174 + r235177;
        double r235179 = r235173 * r235178;
        double r235180 = acos(r235179);
        double r235181 = log1p(r235180);
        double r235182 = expm1(r235181);
        double r235183 = log(r235182);
        double r235184 = exp(r235183);
        double r235185 = log1p(r235184);
        double r235186 = expm1(r235185);
        return r235186;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied flip3--0.5

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{{\left(v \cdot v\right)}^{3} - {1}^{3}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)}}}\right)\]

  4. Applied associate-/r/0.5

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)}\]
  7. Using strategy rm

  8. Applied add-exp-log0.5

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)}}\right)\right)\]
  9. Using strategy rm

  10. Applied expm1-log1p-u0.5

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)\right)}}\right)\right)\]

  11. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)\right)}\right)\right)\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))